Uncertainty principle if position is restricted

Due to the uncertainty principle, it is not possible to have both a small uncertainty in momentum and a finite length for the boxes. This is because the particles cannot leave the boxes, so the finite length serves as an upper limit for the uncertainty in position. In summary, the uncertainty principle prevents us from having both a small uncertainty in momentum and a finite length for the boxes in this scenario.
  • #1
greypilgrim
533
36
Hi.

Assume we have a large number of identical boxes of some finite length ##l## and with infinite potential walls. Let's prepare them all in the same momentum eigenstate. Since for eigenstates ##\Delta p=0##, by the uncertainty principle ##\Delta x## should go to infinity. However, since the particles can't leave the boxes, ##l## is an upper limit for ##\Delta x##. How is this possible?
 
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  • #2
There are no momentum eigenstates of the situation you are describing. They simply do not belong to the appropriate Hilbert space.
 
  • #3
Ok, but can we choose states such that ##\Delta p## is small enough (I guess not, that's probably why this doesn't work)?
 
  • #4
greypilgrim said:
Let's prepare them all in the same momentum eigenstate.
You cannot do that (if other conditions you mentioned are fulfilled).
 
  • #5
greypilgrim said:
Ok, but can we choose states such that ##\Delta p## is small enough (I guess not, that's probably why this doesn't work)?
Exactly!
 

Related to Uncertainty principle if position is restricted

1. What is the uncertainty principle if position is restricted?

The uncertainty principle, also known as the Heisenberg uncertainty principle, states that it is impossible to know the exact position and momentum of a particle at the same time. This means that the more precisely we know the position of a particle, the less we know about its momentum, and vice versa. This principle is a fundamental concept in quantum mechanics and has implications for the behavior of particles at the subatomic level.

2. How does the uncertainty principle apply to restricted position?

The uncertainty principle applies to restricted position in the sense that if the position of a particle is known to be within a certain range, there will be a corresponding uncertainty in its momentum. This means that the more we restrict the position of a particle, the greater the uncertainty in its momentum will be, making it impossible to know both quantities with absolute precision.

3. Can the uncertainty principle be violated if position is restricted?

No, the uncertainty principle cannot be violated even if the position of a particle is restricted. This principle is a fundamental aspect of quantum mechanics and has been extensively tested and proven through experiments. It is a fundamental limitation of our ability to measure the properties of particles at the subatomic level.

4. How does the uncertainty principle affect our understanding of the physical world?

The uncertainty principle has a profound impact on our understanding of the physical world. It challenges the classical notion of determinism, which states that all physical quantities can be known with absolute precision. Instead, the uncertainty principle suggests that there is inherent randomness and unpredictability at the subatomic level, and that our ability to measure and observe particles is limited.

5. Is the uncertainty principle limited to position and momentum?

No, the uncertainty principle is not limited to position and momentum. While these are the two quantities that are most commonly associated with the uncertainty principle, it also applies to other pairs of complementary physical properties such as energy and time, or angular momentum and direction. The principle essentially states that any two complementary properties of a particle cannot be known simultaneously with absolute precision.

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